Extremes and limit theorems for difference of chi-type processes

Albin, Patrik; Hashorva, Enkelejd; Ji, Lanpeng; Ling, Chengxiu

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 349-366

Let $\{{\zeta }_{m,k}^{\left(\kappa \right)}\left(t\right),t\ge 0\},\kappa >0$ be random processes defined as the differences of two independent stationary chi-type processes with $m$ and $k$ degrees of freedom. In this paper we derive the asymptotics of $\mathbb{P}\{{sup}_{t\in [0,T]}{\zeta }_{m,k}^{\left(\kappa \right)}\left(t\right)>u\},u\to \infty$ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.

Classification : 60G15, 60G70
Keywords: Stationary Gaussian process, stationary chi-type process, extremes, Berman sojourn limit theorem, Gumbel limit theorem, Berman’s condition

J.M.P. Albin, On extremal theory for non differentiable stationary processes. Ph.D. thesis, University of Lund, Sweden (1987).

J.M.P. Albin, On extremal theory for stationary processes. Ann. Probab. 18 (1990) 92–128. | Zbl

J.M.P. Albin and D. Jarušková, On a test statistic for linear trend. Extremes 6 (2003) 247–258. | Zbl | DOI

A. Aue, L. Horváth and M. Hušková, Extreme value theory for stochastic integrals of Legendre polynomials. J. Multivariate Anal. 100 (2009) 1029–1043. | Zbl | DOI

S.M. Berman, Sojourns and extremes of stationary processes. Ann. Probab. 10 (1982) 1–46. | Zbl | DOI

S.M. Berman, Sojourns and extremes of stochastic processes. The Wadsworth and Brooks/Cole Statistics/Probability Series. Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, CA (1992). | Zbl

K. Dȩbicki, E. Hashorva and L. Ji, Gaussian approximation of perturbed chi-square risks. Stat. Interface 7 (2014) 363–373. | Zbl | DOI

A.B. Dieker and B. Yakir, On asymptotic constants in the theory of Gaussian processes. Bernoulli 20 (2014) 1600–1619. | Zbl | DOI

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling extremal events for insurance and finance. Springer-Verlag, Berlin (1997). | Zbl

E. Hashorva and L. Ji, Piterbarg theorems for chi-processes with trend. Extremes 18 (2015) 37–64. | Zbl | DOI

E. Hashorva, D. Korshunov and V.I. Piterbarg, Asymptotic expansion of Gaussian chaos via probabilistic approach. Extremes 18 (2015) 315–347. | Zbl | DOI

J. Hüsler, V.I. Piterbarg and O. Seleznjev, On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003) 1615–1653. | Zbl | DOI

D. Jarušková, Detecting non-simultaneous changes in means of vectors. TEST 24 (2015) 681–700. | Zbl | DOI

C. Klüppelberg and M.G. Rasmussen, Outcrossings of safe regions by generalized hyperbolic processes. Stat. Probab. Lett. 83 (2013) 2197–2204. | Zbl | DOI

M.R. Leadbetter, G. Lindgren and H. Rootzén, Vol. 11 of Extremes and related properties of random sequences and processes. Springer Verlag (1983). | Zbl

M.R. Leadbetter and H. Rootzén, Extreme value theory for continuous parameter stationary processes. Z. Wahrsch. Verw. Gebiete 60 (1982) 1–20. | Zbl | DOI

G. Lindgren, Extreme values and crossings for the $\chi {}^2$-process and other functions of multidimensional Gaussian processes, with reliability applications. Adv. Appl. Probab. 12 (1980) 746–774. | Zbl

G. Lindgren, Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes. Stochastic Process. Appl. 17 (1984) 285–312. | Zbl | DOI

G. Lindgren, Slepian models for $\chi {}^2$-processes with dependent components with application to envelope upcrossings. J. Appl. Probab. 26 (1989) 36–49. | Zbl

C. Ling and Z. Tan, On maxima of chi-processes over threshold dependent grids. Statistics 50 (2016) 579–595. | Zbl | DOI

C. Ling and Z. Peng, Extremes of order statistics of self-similar processes (in Chinese). Sci. Sin. Math. 46 (2016) 1–10. DOI: 10.1360/012016-15.

V.I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Vol. 148. American Mathematical Society, Providence, RI (1996). | Zbl

V.I. Piterbarg and A. Zhdanov, On probability of high extremes for product of two independent Gaussian stationary processes. Extremes 18 (2015) 99–108. | Zbl | DOI

O. Seleznjev, Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields. Extremes 8 (2006) 161–169 (2005). | Zbl | DOI

Z. Tan and E. Hashorva, Exact asymptotics and limit theorems for supremum of stationary $\chi$-processes over a random interval. Stochastic Process. Appl. 123 (2013) 1983–2998. | Zbl

Z. Tan and E. Hashorva, Limit theorems for extremes of strongly dependent cyclo-stationary $\chi$-processes. Extremes 16 (2013) 241–254. | Zbl | DOI